Collapsibility of Random Clique Complexes

We prove a sufficient condition for a finite clique complex to collapse to a $k$-dimensional complex, and use this to exhibit thresholds for $(k+1)$-collapsibility in a sparse random clique complex. In particular, if every strongly connected, pure $(k+1)$-dimensional subcomplex of a clique complex $X$ has a vertex of degree at most $2k+1$, then $X$ is $(k+1)$-collapsible. In the random model $X(n,p)$ of clique complexes of an Erd\H{o}s–R'{e}nyi random graph $G(n,p)$, we then show that for any fixed $k\ge 0$, if $p=n^{-\alpha }$ for fixed $1\mathrm{/}(k+1)<\alpha <1\mathrm{/}k$, then a clique complex $XX(n,p)$ is $(k+1)$-collapsible with high probability.